Arctic sea ice extent is governed by an energy balance between incoming solar radiation (modified by surface albedo) and outgoing longwave radiation, combined with the thermodynamic growth law derived from Stefan's equation: h = √(2k·FDD·86400 / (ρ·L)), where h is ice thickness, k = 2.0 W/m/K is the thermal conductivity of sea ice, FDD are cumulative freezing degree-days, ρ = 917 kg/m³ is ice density, and L = 334 kJ/kg is the latent heat of fusion. The most important non-linearity is the ice-albedo feedback: sea ice reflects about 80% of incoming sunlight (α = 0.8) while open ocean absorbs 94% (α = 0.06), so losing ice exposes dark water that absorbs more heat, accelerating further melting — a self-reinforcing positive feedback that can produce abrupt tipping-point transitions.
You can control air temperature, solar radiation, CO₂ radiative forcing (0–8 W/m²), and initial ice thickness. The Arctic view shows ice extent evolving through the annual cycle, and the annual cycle plot reveals the seasonal minimum (September) and maximum (March). Increasing CO₂ forcing progressively reduces the summer minimum until it vanishes — the "ice-free Arctic summer" threshold.
What is the ice-albedo feedback?
Albedo is the fraction of solar radiation a surface reflects. Sea ice has a high albedo of around 0.8 (it reflects 80% of sunlight), while open Arctic Ocean has an albedo of only 0.06. When warming causes ice to melt, it exposes dark ocean water that absorbs far more heat, accelerating further warming and melting in a self-amplifying positive feedback loop — one of the primary reasons the Arctic is warming about four times faster than the global average.
What does Stefan's law predict for ice growth?
Stefan's law predicts that sea ice thickness grows as h = √(2k·(T_f − T_a)·t / (ρ·L)), where T_f = 0°C is the freezing point, T_a is the air temperature, and t is elapsed freezing time. This square-root dependence means ice grows quickly at first but increasingly slowly as the insulating ice layer thickens, making the atmosphere-ocean heat exchange less efficient.
How much CO₂ forcing is needed to eliminate summer ice?
Current IPCC projections suggest the Arctic is likely to experience its first ice-free September before 2050 under SSP2-4.5 (approximately +3–4 W/m² of CO₂ forcing above pre-industrial). The simulation shows this threshold: as you increase CO₂ forcing, September ice extent progressively decreases, and above a critical value the annual cycle never reaches positive thickness — permanent summer ice loss.
Freezing degree-days are the cumulative sum of daily temperatures below 0°C over a season: FDD = Σ max(0, T_f − T_day). They are widely used in sea ice models as a measure of total freezing energy available. A colder or longer winter accumulates more FDD, producing thicker ice. The simulation integrates FDD day by day from a sinusoidal seasonal temperature cycle with amplitude ±15°C.
September marks the annual sea ice minimum in the Arctic, after months of summer melting. Satellite records from NSIDC show September ice extent has declined by roughly 13% per decade since 1979. The September 2012 record minimum of 3.41 million km² remains the lowest ever measured. Scientists focus on September because it is the most sensitive indicator of long-term warming trends.
Models and observations suggest a "tipping point" at which summer ice loss becomes self-sustaining through the albedo feedback. Some studies place this threshold at around 1.5–2°C of global warming above pre-industrial levels, corresponding to a September ice extent below ~1 million km². However, there is debate about whether this tipping point is abrupt or gradual, and whether it is reversible if CO₂ concentrations are subsequently reduced.
Sea ice loss amplifies Arctic warming, reducing the temperature gradient between the Arctic and mid-latitudes. This weakens and destabilises the polar jet stream, potentially increasing the frequency of "blocking" events — persistent atmospheric patterns that cause extreme weather in Europe and North America. The "warm Arctic, cold continents" pattern observed in recent winters may partly reflect this mechanism.
Sea ice is frozen seawater floating on the ocean surface; it is typically 1–3 m thick and reforms each winter. Glaciers and ice sheets (like Greenland and Antarctica) are land-based accumulations of compacted snow, thousands of metres thick, that formed over millennia. Melting sea ice does not directly raise sea level (the ice already displaces water), but melting land ice does. However, sea ice loss accelerates warming through the albedo feedback.
The feedback factor is calculated as 0.8 / α, where α is the current surface albedo. It represents how much more heat the current surface absorbs compared to a fully ice-covered surface. An ice-free ocean (α = 0.06) has a feedback factor of about 13×, meaning it absorbs 13 times more solar energy than ice. This factor quantifies the amplifying role of the albedo feedback in the overall energy balance.
The shortwave (SW) radiation control sets the incoming solar flux in W/m². The energy absorbed by the surface is (1 − α)·SW, so even a moderate increase in SW dramatically increases energy absorption when ice is absent. Real Arctic solar radiation varies from near zero in polar winter to over 400 W/m² at midsummer, and is influenced by cloudiness, aerosols, and the solar cycle.
Arctic sea ice extent responds to the balance between absorbed solar radiation and outgoing heat loss, combined with Stefan's law of ice growth and the powerful ice-albedo feedback: bright ice reflects roughly 80% of sunlight while dark open ocean absorbs about 94%, so each metre of ice lost accelerates further melting. This model tracks ice thickness through a full seasonal cycle and lets you push the system towards an ice-free summer.
A single sea-ice column grows in winter by conductive freezing (Stefan's law) and shrinks in summer as absorbed sunlight depends on the surface's own albedo, which itself depends on whether ice is still present — the feedback loop that makes Arctic warming self-reinforcing.
Set Air temperature Ta (−40°C to 5°C), Solar radiation SW (50–400 W/m²), CO₂ forcing (0–8 W/m²) and Initial ice thickness (0.1–5.0 m), then press Run Annual Cycle to animate ice thickness, extent and albedo through a simulated year.
September 2012 saw the lowest recorded Arctic sea ice extent, 3.41 million km², and satellite records show the September minimum has shrunk by roughly 13% per decade since 1979.
Sea ice has an albedo around 0.8, reflecting most incoming sunlight, whereas open ocean's albedo is only about 0.06. As warming melts ice, darker water is exposed and absorbs far more solar energy, warming the region further and melting more ice — a self-amplifying loop visible in the model's Albedo α and Feedback factor readouts.
Thickness grows as h = √(2k·(Tf−Ta)·t / (ρ·L)), using the thermal conductivity of ice, cumulative freezing time, ice density and the latent heat of fusion. The square-root form means growth is fastest for thin ice and slows as thicker ice increasingly insulates the ocean beneath it.
It adds extra downward radiative forcing, in watts per square metre, representing the warming effect of greenhouse gases on the energy balance. Raising it shifts the seasonal temperature cycle warmer, shrinking the simulated summer ice minimum until, past a threshold, no ice forms even in winter.
Air temperature is driven by a sine wave with amplitude 15°C around Ta, approximating the real Arctic's winter-to-summer swing so that freezing degree-days accumulate realistically across the year and the thickness curve shows a proper spring maximum and autumn minimum.
When simulated ice thickness falls below about 0.1 m, the model switches the surface albedo to the open-ocean value and displays an ice-free warning — a simplified stand-in for the scientific convention of defining an ice-free Arctic as extent below one million square kilometres.